 
Summary: Graphs and Combinatorics 5, 95106 (1989)
Graphsand
Combinatorics
© SpringerVerlag 1989
Legitimate Coloringsof ProjectivePlanes
N. Alon 1. and Z. FiJredi 2
1 Department of Mathematics, Sackler Faculty of ExactSciences, Tel Aviv University, Ramat
Aviv, Tel Aviv, Israel
2 Mathematical Institute of the Hungarian Academy of Sciences, Budapest, P.O.B. 127, H1364,
Hungary
Abstract. For a projective plane Pn of order n, let X(Pn)denote the minimum number k, so that
there is a coloring of the points of P~in k colors such that no two distinct lines contain precisely
the same number' of points of each color. Answering a question of A. Rosa, we show that for all
sufficiently large n, 5 < X(Pn)< 8 for every projective plane P, of order n.
1. Introduction
Let P = Pn = (P, ~) be a projective plane of order n, with a set of points P and a
set of lines ~. As is well known, P has n2 + n + 1 points and n2 + n + 1 lines with
n + 1 points on every line. A gcolorin9 of P is a function f from P to the set
{1, 2..... X}, which may also be viewed as the (ordered)xpartition (P1, P2 ..... Px)
of P defined by Pi = fl(i). Let C be a Xcoloring of P, corresponding to the
